12 UNBOUNDED NORMAL OPERATORS 415 (ii) A closed vector subspace G of E x E is not the graph of a closed operator if and only if, for some x e pr^G), there exist at least two distinct points (x, y^ and (x, y2) belonging to G, or equivalently (since G is a vector subspace) that (0, yl - y2) e G. But to say that dom(T*) is not dense in E means that there exists z *£ 0 in E orthogonal to dom(T*) (6.3.1), or equiv- alently that (z, 0) is orthogonal to r(T*), or again that (0, -z) belongs to F(T). It follows that (0, — z) cannot be contained in the graph of any closed operator which extends T. Conversely, if dom(T*) is dense in E, then T** is defined, and T(T**) is the orthogonal supplement of /(r(T*)); but this ortho- gonal supplement is also equal to - r(T) (6.3.1). We say then that T** is the closure of T. (15.12.5) Given two not necessarily bounded operators U9 V on E, the vector U • x + V • x is defined for all x E dom(£/) n dom(F), and we denote by U + V the linear mapping xh-» U • x + V • x of dom(C7) n dom(F) into E. In particular, if V is everywhere defined, then dom(£/~f- V) = dom(lT), and the graph T(t/+ V) is the image of T(U) under the linear mapping (x, y) h-» (jc, y + V-x)ofE x E into E x E. If V is continuous and U is closed, it follows therefore that U 4- V is closed, since the mapping (x,y)\->(x,y + V-x) and its inverse (x, y) »-> (x, y — V • x) are continuous. Again, the vector U • (V - x) is defined for the set of all x e E such that xedom(F) and V* xe dom(£7). This set is a vector subspace which we denote by dom(£/F), and UV denotes the linear mapping jc>~» U • (V • x) of dom(£/F) into E. If T is a not necessarily bounded operator which is an infective mapping of dom(T) into E, we denote by T"1 the inverse mapping of T(dom(T)) = dom(T~1) into E. The graph HT'1) is the image of T(T) under the mapping (x,y)\-*(y,x). Hence T""1 is closed if T is closed (and injective). (15.12.6) (von Neumann) Let T be a closed operator on E such that dom(7) is dense in E. Then dom(r*r) is dense in E; the operator T*T is closed; and the operator I + T*T (defined on dom(r*7)) is a bijection of dom(r*T) onto E. The operator B = (I+T*T)~1 is defined on E, continuous, self-adjoint and injective, and its spectrum is contained in the interval [0, 1] of R. Furthermore, the hermitian form (x, y))-+(B • x\y) is positive and nondegenerate, and C - TB is a continuous operator defined on E, such that C(E) c: dom(r*). Finally,e restriction of G112